A probe descending through Mar's atmosphere records pressure...

A probe descending through Mar's atmosphere records pressure as a function of altitude; the data are in the table below. Plot the natural logarithm of the pressure versus altitude and fit a line to your plotted points. Mars's atmospheric pressure is governed by the same equation that describes Earth's; see Problem 70. Use your fitted line, in connection with that equation, to determine (a) Mars's surface pressure and (b) the scale height $h_{0}$ $$\begin{array}{|l|c|c|c|c|c|c|} \hline \text { Altitude (km) } & 10 & 20 & 30 & 40 & 50 & 60 \\ \hline \text { Pressure (Pa) } & 242 & 98.7 & 37.6 & 16.2 & 7.21 & 2.38 \\ \hline \end{array}$$

A probe descending through Mar's atmosphere records pressure as a function of altitude; the data are in the table below. Plot the natural logarithm of the pressure versus altitude and fit a line to your plotted points. Mars's atmospheric pressure is governed by the same equation that describes Earth's; see Problem 70. Use your fitted line, in connection with that equation, to determine
(a) Mars's surface pressure and (b) the scale height $h_{0}$
$$\begin{array}{|l|c|c|c|c|c|c|}
\hline \text { Altitude (km) } & 10 & 20 & 30 & 40 & 50 & 60 \\
\hline \text { Pressure (Pa) } & 242 & 98.7 & 37.6 & 16.2 & 7.21 & 2.38 \\
\hline
\end{array}$$

Key Concepts

Surface Pressure Estimation

Estimating the surface pressure involves using the intercept from the linear regression of the logarithmically transformed data. The intercept corresponds to the natural logarithm of the surface pressure, so by exponentiating this value, one obtains the pressure at zero altitude. This concept is crucial for understanding the physical state of the atmosphere at the planetary surface.

Scale Height

The scale height is a parameter that measures how quickly the atmospheric pressure decreases with altitude. It is defined as the altitude at which the pressure falls to 1/e (approximately 37%) of its value at the surface. In the linearized equation obtained via logarithmic transformation, the scale height is mathematically related to the slope of the fitted line.

Linear Regression

Linear regression is the process of fitting a straight line to a set of data points in order to model the relationship between two variables. In this context, it is used on the logarithmically transformed pressure data versus altitude, yielding a line whose intercept and slope can be directly related to physical parameters in the model, such as the surface pressure and the scale height.

Logarithmic Transformation

Taking the natural logarithm of the pressure converts the exponential relationship into a linear one. This transformation simplifies the analysis by allowing a straight-line fit to the data, making it easier to estimate key parameters like the intercept and slope. The transformed equation can then be analyzed using linear regression techniques.

Exponential Atmosphere Model

This concept describes how the pressure in a planetary atmosphere decreases exponentially with altitude. It is based on the idea that the density of the atmosphere falls off with height, which in turn affects the pressure due to the weight of the gas above. The model is expressed mathematically, often involving parameters such as the surface pressure and the scale height.

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